ERIC EJ802695: Cactus: An Introduction to Regression

Hartley Hyde • CQctus.pQges@internode.on.net

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S ir Francis Galton (1822-1911) studied
medicine at Cambridge but when his
father died in 1844 he no longer needed to
work so he embarked on a tour of the Nile.
After some exploration of Namibia he gave up
travelling and settled to a life of science.

As a psychologist he introduced the idea of
a survey to collect data and was the first to
promote the study of twins. His use of maps
to show high air pressure areas led to the
development of scientific weather forecasting.

An experiment with breeding sweet peas
inspired Galton to think up the idea of regres-
sion analysis in the 1870s and of statistical
correlation in 1888.

Using these statistical tools he was able to
convince Scotland Yard of the benefit of using
fingerprints to identify people.

In later years he became one of the first to
apply the evolutionary theories of his cousin
Charles Darwin to human populations.
Having no children, he left most of his fortune
to the University College of London where his
younger colleague Karl Pearson continued the
development of statistics.

condensed from the Wikipedia

An introduction to regression

W hen 1 first used VisiCalc, 1 thought it a
very useful tool when 1 had the formulas,
but how could 1 design a spreadsheet if there
was no known formula for the quantities 1 was
trying to predict? A few months later 1 learned
to use multiple linear regression software and
suddenly it all clicked into place: all 1 needed
was a data sample and the regression software
would give me a formula and some idea of the
limits of accuracy. Spreadsheets and regression
both existed long before computers, but they
became much more powerful tools in their
computer form.

While some topics in mathematics appead to
our sense of elegance there are others, like
regression, that grab our attention because of
their utility. Respect for elegance or utility are
reactions that grow from having understood a
topic, but they do not help to introduce it.

When introducing a new topic, we need a
variety of activities in the hope of catching the
interest of a corresponding variety of learning
styles. 1 try to include something of the history
of the people who first explored the topic. The
story of Sir Francis Gabon’s contribution to
science and statistics leaves little doubt that his
development of statistics arose from many prac-
tical and innovative pursuits.

Students who think visually are often helped
by the demonstration to be found at
WWW. dynamicgeometry . com / J avasketchpad /
gallery /pages /least squares.php. This demon-
stration was developed by Bill Finzer and is
included on the Geometer’s SketchPad site as
one of several examples of how a
JavaSketchPad model can be built into a web
page. The screen dump on the next page has
had to be simplified from the highly coloured,
dynamic version that you can find at the
website.

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Calculator And Computer Technology User Service

The data points are labelled P(l), P(2) ... and
the task is to move the oblique line until the
sum of squares of the distances between the
data points and the line is minimised. The large
square at the bottom right-hand corner has an
area equal to the sum of the areas of the smaller
squares. By moving the coloured dots labelled
slope and y-intercept, the gradient and height of
the regression line can be changed. All you have
to do is keep fiddling until the total area has its
least value. This is not a trivial task.

This activity may be all that some students
will need to develop sufficient confidence in
what is happening when their calculator fits a
regression line to a set of data points. This is
something of a black-box approach in which we
do not know how it works but we do not care
anyway.

Other students need to know how the
demonstration works. The code is all there in
the public domain: Just click on View and select
Source. The instruction set is in the page
header and begins:

{1} Point(359,246)[hidden];

{2} Point(356,35)[label('P(6)')];

{3} Point(311,63)[label('P(5)')];

{4} Point(252,82)[label('P(4)')];

down version on their own graphics calculator.
The following example works well on a ClassPad.

Plot and constrain the points A (-2,4),
B (2,1), C (4,-4) and D (0,0) as shown below.

Plot the points E (3,0) and F(0,4). Construct
and constrain the axes DE and DF and hide E
and F. Check that you now have the non-
dynamic parts firmly fixed in place.

Define a point G on the y-axis DF. Plot a
point H at (-3,3). Draw the future regression
line GH. If you move the point G, you will
change the y-intercept, and you can change the
slope by moving the point H.

Highlight the x-axis DE as well as the point A
and construct a perpendicular line. Point A and
the line DE are constrained, so is the perpen-
dicular line. Place similar lines through B and C
perpendicular to DE. Identify the intersections
of the new lines with GH as the points I, J and
K as shown below. As you move either of the
points G or H you will see that the points I, J
and K are constrained to follow the tramtracks
AT, BJ and CK Select the tramtracks and hide
them. Check that I, J and K are still constrained
to the vertical lines through the points A, B and
C. The line of best fit is found when the points

followed by another 84 lines. Each line is easy
to follow, but the whole construction is complex.
After all, this is a demonstration piece. If your
students have already learned to use
JavaSketchPad, you could let them satisfy their
curiosity by playing with the construction and
making minor alterations so that they can see
what each section of the code is doing. However,
while you may be justified in teaching
JavaSketchPad to a geometry class, you may
not wish to invest that much time with a statis-
tics class. On the other hand, you can
reasonably expect that they might try a cut-

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G and H are moved such that the expression
(AT^ + + CK^) is minimised. To illustrate this

we will build little squares on each of the line
segments AI, BJ and CK.

As mentioned in a previous article, it is not
easy to construct stable quadrilaterals that will
withstand manipulations. In this case we
constrain the angles ML and ATM to be 90° and
set the slope of LM to oo so as to form a
rectangle. Make AI and AL equal, thus forcing
AIML to be a square. Repeat this procedure with
BJON and CKQP. We should now have three
squares that change size as we change the posi-
tions of G and H.

Select the three points A, L and M. At the left-
hand end of the measurement bar, select the
Area icon and, for this exaimple, the area of
ALMI is about 1.01 unit^. Tap on the area
measurement and drag it toward the bottom of
the work area.

This will leave the title “Area:” in the
Measurement Window. You can now edit the
word to a more appropriate description. Just
change it to read “Area A:”. Tap the tick. Then
repeat for the other two squares like this.

From the Draw Menu choose Expression.
Each of the previous measurements is now
numbered in a small box. Tap on the first box
and @1 appears in the Measurement Bar. Type
“+”. Then tap on the second and you have
@l+@2 in the Measurement Bar. Keep going
until you have @l+@2+@3 and then tap the
tick. You now have a total area to slide down
under the other area measurements.

All the students have left to do is move the
point G to different places ailong the y-axis and
move H to different places to change the slope;
they should be able to get the expression for the
to tail area close to 1.75 as shown below. By
choosing the initial three points A, B and C
carefully, 1 have ensured rational coefficients for
the regression line which can be viewed in the
measurement bar.

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You can now show students how to use the
regression software which is built into the
spreadsheet to obtain the same answer much
more quickly.

Simple geometric examples like
this assist visual thinkers to build
a helpful dynamic model of how
the regression line is determined.

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